You are creating a rectangular garden and want to surround it with brick edging. You also are going to divide it into two halves with the edging so that your vegetables and herbs are on one side and flowers are on the other.
You only have 120 ft worth of brick edging and want to enclose the most are possible. Determine the dimensions (Length and Width) that will maximize the total area of the garden, and state that maximum area
you will have 2 strips of length x and 3 strips of length y.
2x+3y = 120
the area a is
a = xy = x(40 - 2/3 x) = 40x - 2/3 x^2
da/dx = 40 - 4/3 x
for max area, da/dx=0, so x = 30
Thus y=20, and the edging is evenly divided between width and length.
max area is thus 600 ft^2
To determine the dimensions that will maximize the total area of the garden, we can use calculus. Let's define the length of the garden as L and the width as W.
Since the brick edging goes all around the garden, we need to consider that the edging will be used for both the length and the width of the garden. So, the total length of the edging used will be 2L + 2W.
We know that the total length of the edging is 120 ft, so we have the equation:
2L + 2W = 120
Now, let's express the total area of the garden in terms of L and W. The area of a rectangle is calculated by multiplying the length and width, so the total area is given by A = L * W.
To find the maximum area, we need to find the maximum of this function. We can do this by using the equation we found earlier to solve for one variable and substitute it into the area function.
From 2L + 2W = 120, we can solve for one variable, say L:
2L = 120 - 2W
L = 60 - W
Now we substitute L into the area function:
A = (60 - W) * W
A = 60W - W^2
To find the maximum area, we need to take the derivative of the area function with respect to W and set it equal to 0:
dA/dW = 60 - 2W = 0
Solving for W:
60 - 2W = 0
2W = 60
W = 30
Now that we have the value for W, we substitute it back into the equation for L:
L = 60 - W
L = 60 - 30
L = 30
Therefore, the dimensions of the garden that will maximize the total area are L = 30 ft and W = 30 ft. The maximum area is obtained by substituting these values into the area function:
A = L * W
A = 30 * 30
A = 900 sq ft
So, the maximum area for the garden is 900 square feet.